Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{7p}{2(3p - 4)} \div \dfrac{7}{18p - 24} $
Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{7p}{2(3p - 4)} \times \dfrac{18p - 24}{7} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ 7p \times (18p - 24) } { 2(3p - 4) \times 7 } $ $ r = \dfrac {7p \times 6(3p - 4)} {7 \times 2(3p - 4)} $ $ r = \dfrac{42p(3p - 4)}{14(3p - 4)} $ We can cancel the $3p - 4$ so long as $3p - 4 \neq 0$ Therefore $p \neq \dfrac{4}{3}$ $r = \dfrac{42p \cancel{(3p - 4})}{14 \cancel{(3p - 4)}} = \dfrac{42p}{14} = 3p $